Confusion about fragments and complete sentences

Broadly speaking, there are three ways to define a sentence: orthographically, notionally and grammatically/formally.

Orthographical: A sentence is a string of words, starting with a capitalised word and ending with a full-stop, question mark or exclamation mark.

Notional: A sentence is a string of words expressing a complete idea and that can stand alone.

Grammatical/Formal: A sentence is a string of words containing a subject and finite verb.

The notional definition is problematic because "a complete idea" is a slippery term. By extension, the statement that "fragments are incomplete sentences and cannot stand alone" is similarly problematic and leads to confusion such as you are experiencing.

A better way to understand why "Not realizing at the time that he needed ginger as well" is a fragment and "To think about this makes me happy" is a sentence is to apply the grammatical definition.

"To think about this makes me happy" contains a subject: To think about this and a finite verb makes. Hence it is a sentence by the grammatical definition.

On the other hand, the word string "Not realizing at the time that he needed ginger as well" does not have a subject and does not have a finite verb. It is not a complete sentence and is hence classified as a fragment.

A full explanation would require me to account for the subject and the finite verb in the subordinate content clause that he needed ginger as well. But this should be sufficient for you to understand why your teacher is right.

Here is an article about finite verbs.